• 沒有找到結果。

The longest detour problem on a shortest path tree

N/A
N/A
Protected

Academic year: 2021

Share "The longest detour problem on a shortest path tree"

Copied!
5
0
0

加載中.... (立即查看全文)

全文

(1)7+( /21*(67 '(7285 352%/(0 21 $ 6+257(67 3$7+ 75((. ½. )X/RQJ <HK 6K\XH0LQJ 7DQJ <XH/L :DQJ DQG 7LQJ<HP +R. 'HSDUWPHQW RI ,QIRUPDWLRQ 0DQDJHPHQW 1DWLRQDO 7DLZDQ 8QLYHUVLW\ RI 6FLHQFH DQG 7HFKQRORJ\ 7DLSHL &LW\ 7DLZDQ 5 2 & (PDLO \OZDQJ#FVQWXVWHGXWZ. G* U V

(2)  LV WKH OHQJWK RI 3* U V

(3)  $%675$&7. /HW 3* U V

(4) UžXYžV! EH WKH VKRUWHVW SDWK IURP U WR V $ GHWRXU DW YHUWH[ X GHQRWHG DV 3*H X V

(5)  LV. ,Q D ELFRQQHFWHG JUDSK D GHWRXU IURP D YHUWH[ X WR VRPH GHVWLQDWLRQ YHUWH[ V LV GHILQHG DV DQ DOWHUQDWLYH VKRUWHVW. GHILQHG DV D VKRUWHVW SDWK IURP X WR V ZLWKRXW XVLQJ WKH. SDWK IURP X WR V ZKHQ WKH HGJH X Y

(6) LV QRW DYDLODEOH LQ D. HGJH H. VKRUWHVW SDWK X Y  V! 7KH ORQJHVW GHWRXU /'

(7). HGJH H IURP * 1RWLFH WKDW D GHWRXU LV D VKRUWHVW SDWK IURP. SUREOHP LV WR ILQG DQ HGJH X Y

(8)  FDOOHG WKH GHWRXUFULWLFDO. X WR V QRW U WR V LQ *H 7KH ORQJHVW GHWRXU /'

(9) SUREOHP. X Y

(10)  7KH JUDSK *H LV REWDLQHG E\ UHPRYLQJ. HGJH DORQJ D VKRUWHVW SDWK UžXYžV! VXFK WKDW WKH. LV WR ILQG DQ HGJH H X Y

(11)  FDOOHG WKH GHWRXUFULWLFDO HGJH. UHPRYDO RI X Y

(12) PD\ FDXVH PD[LPXP LQFUHPHQW RI. LQ 3* U V

(13) VXFK WKDW G*H X V

(14) PLQXV G* X V

(15) LV. GLVWDQFH IURP X WR V 7KH /' SUREOHP FDQ EH VROYHG LQ. PD[LPXP. 2 P  Q ORJ Q

(16) WLPH ZKHUH P DQG Q GHQRWH WKH QXPEHU RI YHUWLFHV DQG HGJHV UHVSHFWLYHO\ LQ D JUDSK ,Q WKLV SDSHU ZH DUH FRQFHUQHG ZLWK WKH /' SUREOHP ZLWK UHVSHFW WR D VKRUWHVW SDWK WUHH RI D JUDSK $Q 2 P. α P. . Q

(17)

(18) WLPH. DOJRULWKP IRU ILQGLQJ D GHWRXUFULWLFDO HGJH LQ D VKRUWHVW SDWK WUHH LV SUHVHQWHG LQ WKLV SDSHU ZKHUH. . α LV D IXQFWLRQDO. . . . . . . . /RQJHVW GHWRXU 'HWRXUFULWLFDO HGJH. . . . . .  ,1752'8&7,21. .  .  . 6KRUWHVW SDWK %LFRQQHFWHG JUDSKV. . . . LQYHUVH RI $FNHUPDQQ©V IXQFWLRQ .H\ZRUGV . . .  . . . . . /HW * 9 (

(19) EH DQ XQGLUHFWHG JUDSK ZKHUH 9 DQG ( DUH YHUWH[ VHW DQG HGJH VHW UHVSHFWLYHO\ $ ZHLJKWHG JUDSK LV D JUDSK LQ ZKLFK HYHU\ HGJH H. ∈. )LJXUH  $ ZHLJKWHG JUDSK *. ( LV DVVRFLDWHG ZLWK D. QRQQHJDWLYH UHDO ZHLJKW Z H

(20) ZKLFK FDQ DOVR EH GHQRWHG. )RU H[DPSOH VHH )LJXUH  3* 

(21). DV Z [ \

(22) LI [ DQG \ DUH WZR HQG YHUWLFHV RI H  7KH OHQJWK.   ! LV WKH. VKRUWHVW SDWK IURP  WR  3DWKV ! DQG ! DUH. RI D SDWK LV WKH ZHLJKW VXPPDWLRQ RI HGJHV LQ WKH SDWK $. WKH GHWRXUV DW YHUWLFHV  DQG  UHVSHFWLYHO\ ZLWK UHVSHFW. VKRUWHVW SDWK EHWZHHQ YHUWLFHV U DQG V LQ * GHQRWHG DV. WR 3* 

(23)  ,Q RWKHU ZRUGV SDWKV ! DQG !. 3* U V

(24)  LV GHILQHG DV D SDWK ZLWK WKH VKRUWHVW OHQJWK IURP. DUH WDNHQ DV WKH VKRUWHVW SDWKV WRZDUG YHUWH[  ZKHQ HGJHV. U WR V 7KH GLVWDQFH EHWZHHQ YHUWLFHV U DQG V GHQRWHG DV. ½. 7KLV ZRUN ZDV VXSSRUWHG E\ WKH 1DWLRQDO 6FLHQFH &RXQFLO 5HSXEOLF RI &KLQD XQGHU &RQWUDFW 16&(. Ä.

(25) 

(26). DQG. 

(27) . UHVSHFWLYHO\. DUH. QRW. DYDLODEOH. FRQWDLQV WKH FRQFOXGLQJ UHPDUNV. 7KH. GHWRXUFULWLFDO HGJH RI 3* 

(28) LV 

(29)  EHFDXVH G* 

(30) 

(31)  G* 

(32).   . 

(33) 

(34)  G* 

(35).   .  LV JUHDWHU WKDQ G* . . 127$7,21. DV. %HIRUH GHVFULELQJ RXU DOJRULWKP ZH GHILQH VRPH QRWDWLRQ. GHWHUPLQLQJ WKH GHWRXUFULWLFDO HGJH LV LPSRUWDQW IURP WKH. ZKLFK ZLOO EH XVHG WKURXJKRXW WKLV SDSHU 0RVW RI WKHP. YLHZSRLQW RI QHWZRUN PDQDJHPHQW 'XH WR D VXGGHQ OLQN. DUH DOVR XVHG LQ >@. 7KH DOJRULWKP IRU ILQGLQJ GHWRXUV DV ZHOO. /HW 6* V

(36) EH D VKRUWHVW SDWK WUHH RI * URRWHG DW. IDLOXUH IURP VRPH QRGH D PHVVDJH PXVW EH WUDQVPLWWHG WKURXJK D GHWRXU IURP WKH QRGH LQVWHDG RI WKH IDLOHG OLQN. YHUWH[ V $Q HGJH LQ 6* V

(37) LV FDOOHG D WUHH HGJH DQG D. ,Q >@ 1DUGHOOL HW DO JDYH DQ 2 P  Q ORJ Q

(38) DOJRULWKP IRU. QRQWUHH HGJH LI LW LV QRW D WUHH HGJH LQ 6* V

(39)  &OHDUO\ DQ\. ILQGLQJ D GHWRXUFULWLFDO HGJH RQ D VKRUWHVW SDWK ZKHUH P. GHWRXU IURP D YHUWH[ LQ WKH JUDSK PXVW PDNH XVH RI VRPH. DQG Q DUH WKH QXPEHU RI YHUWLFHV DQG HGJHV UHVSHFWLYHO\. QRQWUHH HGJH LQ RUGHU WR DUULYH DW V DJDLQ )RU H[DPSOH. RI D JUDSK. )LJXUH  LV 6* 

(40) RI WKH JUDSK LQ )LJXUH  1RQWUHH HGJH  

(41) LV LQFOXGHG LQ WKH GHWRXU IURP YHUWH[  ZKHQ WUHH. /HW 6* V

(42) EH D WUHH RI WKH VKRUWHVW SDWKV IURP YHUWH[. HGJH  

(43) LV QRW DYDLODEOH. V WR DOO RWKHU YHUWLFHV LQ D JUDSK * 7KH /' SUREOHP ZLWK. UHVSHFW WR 6* V

(44)  FDOOHG WKH 7UHH /RQJHVW 'HWRXU 7/'

(45) SUREOHP LV WR ILQG D GHWRXUFULWLFDO HGJH XY

(46) LQ 6* V

(47) . WUHH HGJH. ZLWK Y FORVHU WR V WKDQ X VXFK WKDW WKH UHPRYDO RI XY

(48). QRQWUHH HGJH. UHVXOWV LQ PD[LPXP LQFUHPHQW RI GLVWDQFH IURP X WR V %\. . DSSO\LQJ 1DUGHOOL©V DOJRULWKP IRU HYHU\ SDWK LQ 6* V

(49)  WKLV SUREOHP FDQ EH VROYHG LQ 2 PQ  Q ORJ Q

(50) WLPH :H VKDOO GHVLJQ DQ 2 P. α PQ

(51)

(52). SUREOHP ZKHUH. α. . WLPH DOJRULWKP WR FRQTXHU WKLV. . . . . . . . . . . . . LV D IXQFWLRQDO LQYHUVH RI $FNHUPDQQ©V. . . IXQFWLRQ. . . .  . ,Q WKH SDVW WKH VKRUWHVW SDWK UHODWHG SUREOHPV KDYH EHHQ VWXGLHG ZLGHO\ >@ 7KH PRVW YLWDO HGJH. . 09(

(53) SUREOHP RU WKH 09( SUREOHP ZKLFK LV D. . . . .  . . . YDULDWLRQ RI WKH VKRUWHVW SDWK SUREOHP KDV DOVR EHHQ VWXGLHG. ZLGHO\. >@. 7KHUH. DUH. WZR. )LJXUH  6* 

(54) RI JUDSK *. GHILQLWLRQV. FRQFHUQLQJ WKH 09( SUREOHP ,Q >@ D PRVW YLWDO HGJH ZLWK UHVSHFW WR WKH PLQLPXP VSDQQLQJ WUHH RI D JUDSK LV. ,W IROORZV WKURXJKRXW WKLV SDSHU WKDW DQ HGJH X Y

(55). WKH HGJH WKDW ZKHQ UHPRYHG UHVXOWV LQ WKH JUHDWHVW ZHLJKW. LQ 6* V

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

(57) LV WKH VDPH. HGJH ZLWK UHVSHFW WR 6* V

(58) . DV WKH PRVW YLWDO HGJH RI 3* V U

(59)  +RZHYHU WKH GHWRXU. 3URRI  /HW [ \

(60) EH WKH ILUVW QRQWUHH HGJH HQFRXQWHUHG LQ. FULWLFDO HGJH RI 3* U V

(61) LV QRW QHFHVVDULO\ WKH VDPH DV WKH. WKH GHWRXU IURP X WR V &OHDUO\ WKH SDWK IURP \ WR V LQ. GHWRXUFULWLFDO HGJH RI 3* V U

(62)  UHPDLQGHU. RI. WKLV. SDSHU. LV. RUJDQL]HG. 6* V

(63) LV D VKRUWHVW SDWK EHWZHHQ \ DQG V ,I WKHUH H[LVWV. DV. DQRWKHU QRQWUHH HGJH [© \©

(64) LQ WKH GHWRXU WKHQ WKH SDWK. IROORZV ,Q 6HFWLRQ  ZH LQWURGXFH VRPH QRWDWLRQ XVHG LQ WKLV SDSHU ,Q 6HFWLRQ  ZH SURSRVH DQ 2 P DOJRULWKP. IRU. VROYLQJ. WKH. 7/'. α PQ

(65)

(66). SUREOHP. IURP \ WR V ZKLFK SDVV WKURXJK [© \©

(67) PXVW EH ORQJHU WKDQ. WLPH. 6HFWLRQ. X Y

(68) EH DQ HGJH LQ 6* V

(69) DQG X EH D. GHWRXU VWDUWLQJ YHUWH[ 7KHUH H[LVWV D GHWRXU IURP X WR WKH. GLIIHUHQW IURP WKH PRVW YLWDO HGJH ZLWK UHVSHFW WR D. 7KH. /HW H. RU HTXDO WR WKH SDWK IURP \ WR V LQ 6* V

(70)  7KXV WKH OHPPD. . IROORZV. Å. 4 ( '.

(71) 6LQFH G* V [

(72). $FFRUGLQJ WR /HPPD  ZH FDOO WKH RQO\ QRQWUHH. G* V X

(73)  G* X [

(74)  WKH .YDOXH RI & [. HGJH LQ D GHWRXU WKH FURVVLQJ HGJH RI WKH GHWRXU 1RWH WKDW. \

(75) FDQ EH GHULYHG DV IROORZV . WKH GHWRXU IURP D YHUWH[ PD\ EH PRUH WKDQ RQH ,Q FDVH RI. . [ \

(76). G* V X

(77)  G* X [

(78)  Z [ \

(79)  G* \ V

(80) . PXOWLSOH GHWRXUV WKHLU OHQJWK PXVW EH HTXDO 7KH OHQJWK RI. ,Q WKH DERYH IRUPXOD G* V X

(81) LV WKH FRPPRQ LWHP RI WKH. D GHWRXU IURP X WR V LV IRUPXODWHG DV  G*H X V

(82). G* V [

(83)  Z [ \

(84)  G* \ V

(85). .YDOXHV RI DOO IXQGDPHQWDO F\FOHV LQ )H ,I WKH .YDOXH RI. G*H X [

(86)  Z [ \

(87)  G*H \ V

(88) . ZKHUH H X Y

(89) LV D WUHH HGJH LQ 6* V

(90) DQG [ \

(91) LV WKH. & [ \

(92) LV PLQLPXP DPRQJ DOO IXQGDPHQWDO F\FOHV LQ )H . FURVVLQJ HGJH RI YHUWH[ X ,Q IDFW G*H X [

(93). WKHQ G* X [

(94)  Z [ \

(95)  G* \ V

(96) PXVW EH PLQLPXP DQG. G* X [

(97) . [ \

(98) LV WKH FURVVLQJ HGJH RI D GHWRXU IURP X. VLQFH [ EHORQJV WR WKH VXEWUHH RI 6* V

(99) URRWHG DW X G* \ V

(100)  VLQFH \ EHORQJV WR WKH VHW. &RQYHUVHO\ VXSSRVH WKDW [ \

(101) LV D FURVVLQJ HGJH. RI YHUWLFHV UHDFKDEOH IURP V ZLWKRXW SDVVLQJ WKURXJK HGJH. RI YHUWH[ X 7KH YDOXH RI G* X [

(102)  Z [ \

(103)  G* \ V

(104) LV. H LQ 6* V

(105)  7KXV ZH KDYH. WKH OHQJWK RI D GHWRXU IURP X WR V ,W PXVW EH PLQLPXP. 0RUHRYHU G*H \ V

(106). G*H XV

(107). DPRQJ DOO IHDVLEOH SDWKV IURP X WR V LQ *  H 7KHUHIRUH. G* X[

(108)  Z [\

(109)  G* \V

(110) . /HW ] EH WKH ORZHVW FRPPRQ DQFHVWRU RI WKH WZR YHUWLFHV. WKH. .YDOXH. RI. & [\

(111). RI D QRQWUHH HGJH [ \

(112)  $ IXQGDPHQWDO F\FOH GHQRWHG DV. IXQGDPHQWDO F\FOHV LQ )H . LV. DOVR. PLQLPXP. DPRQJ. DOO. 4 ( '. & [ \

(113)  LV WKH F\FOH ZKLFK FRQVLVWV H[DFWO\ RI 3* ] [

(114)  [. :H XVH DQ H[DPSOH WR LOOXVWUDWH /HPPD  ,Q. \

(115) DQG 3* \ ]

(116)  7KH .YDOXH RI & [ \

(117)  GHQRWHG DV . [. )LJXUH  IXQGDPHQWDO F\FOHV ZKLFK FRYHU HGJH 

(118) LQ. \

(119)  LV WKH WRWDO OHQJWK RI D FORVH ZDON IURP V WR [ WKHQ WR \. 6* 

(120) LH ) 

(121)

(122) DUH & 

(123)  & 

(124)  & 

(125) DQG & 

(126) . ILQDOO\ IURP \ WR V 7KDW LV. & 

(127) LV D GHWRXU F\FOH RI YHUWH[  EHFDXVH . 

(128) LV WKH. . [ \

(129). VPDOOHVW DPRQJ . 

(130)  . 

(131)  . 

(132) DQG . 

(133) . G* V [

(134)  Z [ \

(135)  G* \ V

(136) . G*  

(137) . /HPPD  VKRZV WKDW ZH FDQ FRPSXWH WKH GHWRXU. G*  

(138)  Z . OHQJWK IURP WKH .YDOXH RI D GHWRXU F\FOH 7KDW LV WKH.  1RWLFH WKDW WKH ORZHVW FRPPRQ. GHWRXU OHQJWK IURP D YHUWH[ X HTXDOV G* X [

(139)  Z [ \

(140) . )RU H[DPSOH VHH )LJXUH  DJDLQ .  

(141) Z  

(142)  G*  

(143) 

(144)  G*  

(145).  DQG. .  

(146). G* \ V

(147). DQFHVWRU RI WKH WZR YHUWLFHV RI D QRQWUHH HGJH [ \

(148) PD\. . [ \

(149)  G* V X

(150)  ZKHUH [ \

(151) LV D FURVVLQJ. EH WKH URRW RI 6* V

(152)  ,Q FDVH RI LGHQWLW\ . [ \

(153) LV WKH WRWDO. HGJH RI YHUWH[ X /HW , X

(154) EH WKH GLVWDQFH LQFUHPHQW IURP. OHQJWK RI IXQGDPHQWDO F\FOH & [ \

(155) . D GHWRXU VWDUWLQJ YHUWH[ X WR WKH GHVWLQDWLRQ YHUWH[ V 7R RQHWRRQH. ILQG D GHWRXUFULWLFDO HGJH H XY

(156) RI 6* V

(157) LV WR ILQG WKH. FRUUHVSRQGLQJ WR D QRQWUHH HGJH PXOWLSOH IXQGDPHQWDO. PD[LPXP , X

(158) DPRQJ DOO GHWRXU VWDUWLQJ YHUWLFHV X LQ. $OWKRXJK. D. IXQGDPHQWDO. F\FOH. LV. F\FOHV PD\ FRYHU WKH VDPH WUHH HGJHV RI 6* V

(159)  )RU. 6* V

(160)  /HPPD  SURYLGHV D IRUPXOD WR FRPSXWH , X

(161). H[DPSOH WUHH HGJH  

(162) LQ )LJXUH  LV FRYHUHG E\ IRXU. HIILFLHQWO\. IXQGDPHQWDO F\FOHV LH &  

(163)  &  

(164)  &  

(165) DQG. /HPPD . /HW X EH D GHWRXU VWDUWLQJ YHUWH[ LQ 6* V

(166) . &  

(167)  ,W LV REYLRXVO\ WUXH WKDW LI * LV ELFRQQHFWHG WKHQ. 7KHQ , X

(168). . [\

(169)   G* VX

(170)  ZKHUH [\

(171) LV WKH FURVVLQJ. HDFK WUHH HGJH RI 6* V

(172) PXVW EH FRYHUHG E\ DW OHDVW RQH. HGJH RI D GHWRXU IURP YHUWH[ X. IXQGDPHQWDO F\FOH /HW X Y

(173) EH WKH XQDYDLODEOH HGJH LQ. 3URRI . 6* V

(174)  DQG [ \

(175) EH D FURVVLQJ HGJH RI YHUWH[ X 5HFDOO. , X

(176). /HW H. XY

(177) EH D WUHH HGJH LQ 6* V

(178) . G*H X V

(179)  G* X V

(180). WKDW D FURVVLQJ HGJH LV WKH RQO\ QRQWUHH HGJH LQ D GHWRXU. G* X [

(181)  Z [ \

(182)  G* \ V

(183)  G* V X

(184). 7KH IXQGDPHQWDO F\FOH & [ \

(185) ZKLFK FRYHUV ERWK [ \

(186). . [ \

(187)  G* V X

(188)  G* V X

(189). DQG X Y

(190) LV FDOOHG D GHWRXU F\FOH RI YHUWH[ X ,Q )LJXUH . . [ \

(191)   G* V X

(192) . &  

(193) LV D GHWRXU F\FOH RI YHUWH[  LQ 6* 

(194)  /HPPD . /HW )H EH WKH VHW RI IXQGDPHQWDO F\FOHV ZKLFK. FRYHU HGJH H LQ 6* V

(195) . & [ \

(196) LV D GHWRXU F\FOH RI H LI. DQG RQO\ LI WKH .YDOXH RI & [ \

(197) LV PLQLPXP DPRQJ DOO IXQGDPHQWDO F\FOHV LQ )H  3URRI . /HW [ \

(198) EH WKH FURVVLQJ HGJH RI D IXQGDPHQWDO. F\FOH & [ \

(199). ∈. )H ZKHUH H. X Y

(200)  DQG DVVXPH ZLWKRXW. ORVV RI JHQHUDOLW\ WKDW X LV LQ WKH VKRUWHVW SDWK IURP V WR [. Æ. 4 ( '.

(201) 6LQFH YHUWH[  LV FRYHUHG E\ & 

(202)  WKHUH LV D.  $1 ()),&,(17 $/*25,7+0 )25 62/9,1* 7+( 7/' 352%/(0. GLUHFWHG HGJH IURP WKH VRXUFH QRGH  WR WKH VLQN QRGH. 7R GHWHUPLQH D GHWRXUFULWLFDO HGJH RI 6* V

(203)  ZH VKRXOG. & 

(204)  $OO RI WKH YHUWLFHV   DQG  DUH FRYHUHG E\ DOO RI. FRPSXWH WKH GLVWDQFH LQFUHPHQW , X

(205) IRU HYHU\ YHUWH[ X LQ. WKH. 6* V

(206) H[FHSW YHUWH[ V ,QWXLWLYHO\ VLQFH WKHUH DUH DW PRVW. 7KHUHIRUH WKHUH H[LVWV D FRPPRQ LQWHUPHGLDWH QRGH LQ WKH. 2 P

(207) IXQGDPHQWDO F\FOHV ZKLFK PD\ FRYHU D GHWRXU. SDWKV IURP WKH VRXUFH QRGHV WR WKH VLQN QRGHV. IXQGDPHQWDO. F\FOHV. & 

(208) . & 

(209). ,Q >@ 7DUMDQ KDV GHVFULEHG DQ 2 P. VWDUWLQJ YHUWH[ ZLWK UHVSHFW WR 6* V

(210)  WKH FRPSXWDWLRQ RI. DQG. & 

(211) . α PQ

(212)

(213). WLPH. , X

(214) QHHGV 2 P

(215) WLPH IRU YHUWH[ X ,W UHTXLUHV 2 PQ

(216) WLPH. DOJRULWKP WR FRQVWUXFW D WUDQVPXWHU IRU D VKRUWHVW SDWK WUHH. WR FRPSXWH , X

(217) IRU DOO WKH YHUWLFHV LQ 6* V

(218)  7KHUHIRUH LW. ZLWK UHVSHFW WR D JUDSK RI VL]H P DQG RUGHU Q 0HDQZKLOH. WDNHV 2 PQ

(219) WLPH WR GHWHUPLQH D GHWRXUFULWLFDO HGJH RI. LW KDV EHHQ SURYHG WKDW D WUDQVPXWHU KDV 2 P. 6* V

(220) E\ XVLQJ WKH QDLYH DOJRULWKP +RZHYHU LQ WKH. QRGHV DQG HGJHV *LYHQ D WUDQVPXWHU ZH FDQ HIILFLHQWO\. IROORZLQJ ZH SURSRVH DQ 2 P. GHWHUPLQH D GHWRXU F\FOH DV ZHOO DV D FURVVLQJ HGJH IRU. α P. Q

(221)

(222) DOJRULWKP WR. VROYH WKLV SUREOHP. HDFK YHUWH[ LQ 6* V

(223) . 7R GHVLJQ DQ HIILFLHQW DOJRULWKP IRU VROYLQJ WKH 7/'. SUREOHP. α PQ

(224)

(225). D. YHUWH[ X LQ 6* V

(226) E\ XVLQJ WKH WUDQVPXWHU RI 6* V

(227)  /HW. WUDQVPXWHU ZKLFK ZDV LQWURGXFHG LQ >@ $ WUDQVPXWHU LV. HDFK QRGH [ RI WKH WUDQVPXWHU KDYH DQ DVVRFLDWHG YDOXH. D. $ [

(228) . GLUHFWHG. ZH. QHHG. DF\FOLF. JUDSK. D. GDWD. WKDW. VWUXFWXUH. UHSUHVHQWV. FDOOHG. :H FDQ FRPSXWH , X

(229) IRU HYHU\ GHWRXU VWDUWLQJ. WKH. UHODWLRQ. ,I [ LV D VLQN QRGH WKHQ $ [

(230) LV HTXDO WR WKH .. EHWZHHQ WKH GHWRXU VWDUWLQJ YHUWLFHV DQG WKH IXQGDPHQWDO. YDOXH RI WKH IXQGDPHQWDO F\FOH [ RWKHUZLVH $ [

(231). F\FOHV LQ 6* V

(232)  ,Q D WUDQVPXWHU RQH VRXUFH QRGH RI LQ. 0LQ\∈1 [

(233) ^$ \

(234) `. GHJUHH ]HUR

(235) UHSUHVHQWV D GHWRXU VWDUWLQJ YHUWH[ LQ 6* V

(236) . VXFFHVVRUV RI [ ,Q )LJXUH  7KH YDOXH EHVLGH HDFK QRGH LV. RQH. WKH DVVRFLDWHG YDOXH RI WKDW QRGH )RU D VRXUFH QRGH X. VLQN. QRGH. RI. RXWGHJUHH. ]HUR

(237). UHSUHVHQWV. D. ZKHUH. 1 [

(238). LV. WKH. LPPHGLDWH. IXQGDPHQWDO F\FOH LQ 6* V

(239)  DQG HYHU\ LQWHUPHGLDWH QRGH. $ X

(240) LV WKH .YDOXH RI D GHWRXU F\FOH RI X 7KXV , X

(241). KDV DW OHDVW WZR RXWGHJUHHV DV ZHOO DV DW OHDVW WZR LQ. $ X

(242)   G* XV

(243) IRU HYHU\ GHWRXU VWDUWLQJ YHUWH[ X LQ. GHJUHHV. 6* V

(244) . :H. ODEHO. FRUUHVSRQGLQJ. YHUWH[. HDFK DQG. VRXUFH HDFK. QRGH. VLQN. ZLWK. QRGH. ZLWK. LWV LWV. 1RZ ZH DUH LQ D SRVLWLRQ WR GHVFULEH RXU DOJRULWKP. FRUUHVSRQGLQJ QRQWUHH HGJH RI 6* V

(245)  7KH IXQGDPHQWDO. IRU VROYLQJ WKH 7/' SUREOHP. SURSHUWLHV RI D WUDQVPXWHU DUH DV IROORZV L

(246) 7KHUH LV D GLUHFWHG HGJH IURP D VRXUFH QRGH X WR D VLQN QRGH & [ \

(247). $OJRULWKP )LQGB'&(. LQ WKH WUDQVPXWHU LI DQG RQO\ LI YHUWH[ X LV FRYHUHG E\. ,QSXW. IXQGDPHQWDO F\FOH & [ \

(248) LQ * LL

(249) :KHQ WZR RU PRUH. ELFRQQHFWHG JUDSK * 9(

(250) . GHWRXU VWDUWLQJ YHUWLFHV VKDUH WZR RU PRUH IXQGDPHQWDO. 2XWSXW. F\FOHV WKHUH H[LVWV D FRPPRQ LQWHUPHGLDWH QRGH LQ WKH. 0HWKRG. SDWKV. 6WHS . IURP. WKH. FRUUHVSRQGLQJ. VRXUFH. QRGHV. WR. WKH. FRUUHVSRQGLQJ VLQN QRGHV )RU H[DPSOH WKH WUDQVPXWHU. $ VKRUWHVW SDWK WUHH URRWHG DW YHUWH[ V 6* V

(251)  RI D. $ GHWRXUFULWLFDO HGJH RI 6* V

(252) . )RU HYHU\ QRQWUHH HGJH [ \

(253) ∈ 6* V

(254)  FRPSXWH WKH. .YDOXH RI & [ \

(255)  ZKHUH. FRUUHVSRQGLQJ WR 6* 

(256) LV VKRZQ DV IROORZV. . [ \

(257). . . . . . . . . . . . . . . . . . . . . . . G* V [

(258)  Z [ \

(259)  G* \ V

(260) . 6WHS . &RQVWUXFW D WUDQVPXWHU RI 6* V

(261) . 6WHS . 2EWDLQ WKH DVVRFLDWHG YDOXH RI $ [

(262) IRU HDFK QRGH. [ LQ WKH WUDQVPXWHU 6WHS . . . )RU HDFK VRXUFH QRGH X LQ WKH WUDQVPXWHU RPSXWH. , X

(263) 6WHS . $ X

(264)   G* X V

(265) . )LQG D GHWRXUFULWLFDO HGJH X Y

(266) RI 6* V

(267)  ZKHUH. , X

(268) LV PD[LPXP DPRQJ DOO WKH DVVRFLDWHG YDOXHV RI & 

(269). . & 

(270). . & 

(271). & 

(272). & 

(273). & 

(274). . . . . & 

(275). WKH VRXUFH QRGHV LQ WKH WUDQVPXWHU. . (QG RI $OJRULWKP )LQGB'&(. )LJXUH  7UDQVPXWHU IRU 6* 

(276) . Ç.

(277) 5HIHUHQFHV. :H DOVR XVH )LJXUH  DV DQ H[DPSOH WR LOOXVWUDWH $OJRULWKP )LQGB'&(. >@. 7KH UHVXOW RI 6WHSV  WR  DUH. 7KHRU\ DQG 3UDFWLFH 3UHQWLFH+DOO ,QF  SS . VKRZQ DV )LJXUH  6WHS  FRPSXWHV , X

(278) IRU HDFK YHUWH[ X :H FDQ VHH WKDW , 

(279). $ 

(280)   G*  

(281). %UDV * %UDVVDUG DQG 3 %UDWOH\ $OJRULWKPLFV. .   . >@. + : &RUOH\ DQG ' < 6KD 0RVW YLWDO OLQNV DQG.  7KH YDOXHV RI , 

(282)  , 

(283) ž , 

(284) DUH   . QRGHV LQ ZHLJKWHG QHWZRUNV 2SHUDWLRQV 5HVHDUFK.       DQG  7KHUHIRUH WKH GHWRXU. /HWWHUV 9RO 1R  SS . FULWLFDO HGJH RI 6* 

(285) LV HGJH 

(286) VLQFH , 

(287). . >@. (. :. 'LMNVWUD. $. QRWH. RQ. WZR. SUREOHPV. LQ. PDNHV WKH ODUJHVW GLVWDQFH LQFUHPHQW ZKHQ HGJH 

(288) LV. FRQQHFWLRQ ZLWK JUDSKV 1XPHULF 0DWKHPDWLFV . XQDYDLODEOH.  SS . 6WHS  WDNHV 2 P

(289) WLPH WR FRPSXWH . [\

(290) IRU DOO RI WKH QRQWUHH HGJHV LQ 6* V

(291)  6WHS  UHTXLUHV 2 P WLPH. WR. FRQVWUXFW. WKH. WUDQVPXWHU. RI. 6* V

(292). 7DUMDQ©V DOJRULWKP 6WHS  DOVR WDNHV 2 P. >@. α PQ

(293)

(294). E\. 0 / )UHGPDQ DQG 5 ( 7DUMDQ )LERQDFFL KHDSV DQG. XVLQJ. WKHLU. XVHV. LQ. LPSURYHG. QHWZRUN. RSWLPL]DWLRQ. DOJRULWKPV -RXUQDO RI WKH $&0 9RO   SS. α PQ

(295)

(296) WLPH WR. . REWDLQ WKH DVVRFLDWHG YDOXHV RI DOO WKH QRGHV LQ WKH. >@. /+ +VX HW DO )LQGLQJ WKH PRVW YLWDO HGJH ZLWK. WUDQVPXWHU 2EYLRXVO\ ERWK 6WHSV  DQG  UHTXLUH 2 Q

(297). UHVSHFW WR PLQLPXP VSDQQLQJ WUHH LQ ZHLJKWHG JUDSKV. WLPH. ,QIRUPDWLRQ 3URFHVVLQJ /HWWHUV   SS . 7KHUHIRUH. )LQGB'&(. LV 2 P. WKH. WLPH. FRPSOH[LW\. RI. $OJRULWKP. α PQ

(298)

(299) . . %\ VXPPDUL]LQJ DERYH GHVFULSWLRQ ZH KDYH WKH. >@. IROORZLQJ WKHRUHP 7KHRUHP. . (. DOJRULWKP. $OJRULWKP )LQGB'&( FDQ VROYH WKH 7/'. SUREOHP LQ 2 P. '. .QXWK. $. JHQHUDOL]DWLRQ. RI. 'LMNVWUD. ,QIRUPDWLRQ 3URFHVVLQJ /HWWHUV  . SS. α P Q

(300)

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¦RQHWRRQH§ IDVKLRQ WKH 7/' SUREOHP ZLOO EH D ¦PDQ\WRRQH§ IDVKLRQ 7KH IRUPHU FDQ EH YLHZHG DV D VSHFLDO FDVH RI WKH ODWWHU ,Q WKH QHDU IXWXUH ZH VKDOO IRFXV RXU VWXG\ RQ DQ HIILFLHQW DOJRULWKP IRU ILQGLQJ D GHWRXUFULWLFDO HGJH ZLWK UHVSHFW WR DOO SDLUHG VKRUWHVW SDWKV LQ DQ XQGLUHFWHG JUDSK 7KDW SUREOHP LV LQ D IDVKLRQ. RI. ¦PDQ\WRPDQ\§. $. SDUDOOHO. DOJRULWKP. GHGLFDWHG WR WKH 7/' SUREOHP LV DQRWKHU WRSLF WKDW QHHGV RXU HIIRUW. È.

(302)

參考文獻

相關文件

Output : For each test case, output the maximum distance increment caused by the detour-critical edge of the given shortest path in one line.... We use A[i] to denote the ith element

• The  ArrayList class is an example of a  collection class. • Starting with version 5.0, Java has added a  new kind of for loop called a for each

Neste trimestre, registou-se, em comparação com o 4º trimestre de 2001, um aumento de 55,2% no valor da importação de Querosene comum (Querosene para uso na iluminação), enquanto

Neste trimestre, registaram-se aumentos de 45,8% e 55,1% no valor das importações de Querosene comum (Querosene para uso na iluminação) e de Fuelóleo, respectivamente; enquanto que

Neste trimestre, registou-se, em comparação com o 1º trimestre de 2001, um aumento de 27,6% no valor da importação de Fuelóleo, enquanto que o valor da importação de Querosene

In the fourth quarter of 2002, the total import value of liquid and gas fuels reached the amount of 339 millions Patacas, representing an increase of 1.2% over the same period of

The results contain the conditions of a perfect conversion, the best strategy for converting 2D into prisms or pyramids under the best or worth circumstance, and a strategy

1.本系為全師培學系,但經本入學管道錄取者為外